There are 4! = 24 ways to rank four objects. However, a friend told me that if ties
are allowed, the number increases to 75.

I
attempted to list all the possibilities by first listing the 24 orderings of
four objects, then using brackets to group ties involving two players, then
group ties involving three players, and finally the single case in which all
four objects are tied. But something has gone wrong; my list includes just 69
possibilities, not 75.

What
happened? Did I miss something, or was my friend mistaken?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesAttend to precision.Make sense of problems and persevere in solving them.CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6

The length and width of a rectangle are whole numbers of centimeters. Neither is
divisible by 6. The area of the rectangle is 36 square centimeters.
What is the perimeter of the rectangle, in centimeters?

ProblemsGrades: 3rd to 5th, 9th to 12th, 6th to 8thMeasurement & DataMathematical PracticesAlgebraic ThinkingSolve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.Make sense of problems and persevere in solving them.Gain familiarity with factors and multiples.Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.Geometric measurement: understand concepts of area and relate area to multiplication and to addition.3.MD.C.7b, 3.MD.D.8, 4.OA.B.4, CCSS.Math.Practice.MP1, 4.MD.A.3

A calendar year is typically referred to as a four‑digit number, as in 2008, or
as a two‑digit number, as in ’08. Sometimes, the two‑digit number divides
evenly into the four‑digit number, with no remainder.

How many times did this happen during the twentieth century?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesAlgebraic ThinkingThe Number SystemNum & Ops Base TenAttend to precision.Make sense of problems and persevere in solving them.Multiply and divide within 100.Compute fluently with multi-digit numbers and find common factors and multiples.Perform operations with multi-digit whole numbers and with decimals to hundredths.5.NBT.B.6, 6.NS.B.2, 3.OA.C.7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6

Would you rather work seven days at $20 per day or be paid $2 the first day and have your salary double every day for a week?

ProblemsGrades: 9th to 12th, 3rd to 5th, 6th to 8thFunctionsNum & Ops Base TenMathematical PracticesAlgebraic ThinkingInterpreting FunctionsGeneralize place value understanding for multi-digit whole numbers.Look for and make use of structure.Reason abstractly and quantitatively.Make sense of problems and persevere in solving them.Solve problems involving the four operations, and identify and explain patterns in arithmetic.3.OA.D.9, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP7, 4.NBT.A.2, HSF-IF.A.3

There are 5 houses on a street: house A, B, C, D and E.
The distance between any two adjacent houses is 100 feet. There are 2 children living in house A, 3
children living in house B, 4 children living in house C, 5 children living in
house D and 6 children living in house E.
If the school bus can only make one stop on that street, in front of
which house should the bus stop so that the sum of walking distance among all
children will be the least?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesAttend to precision.Reason abstractly and quantitatively.Make sense of problems and persevere in solving them.CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6

Write the number 41 in a box. Now move in a counterclockwise direction, creating new
boxes and each time adding 1 to the number inside. This spiral starts out as
follows.

Note
that the numbers in bold — 41, 43, and 47 — are prime numbers
(numbers whose only divisors are themselves and 1), and they occur along a diagonal.
If you keep filling in the spiral, what is the first number that is not prime to
appear along this diagonal?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesAlgebraic ThinkingMake sense of problems and persevere in solving them.Gain familiarity with factors and multiples.4.OA.B.4, CCSS.Math.Practice.MP1

The letters of EAT can be
rearranged to become TEA by moving the third letter (T) to the first
position and by moving the other letters one position to the right. This
process could be described as 1→2, 2→3, 3→1.
When this same process is applied again, then TEA becomes ATE.

Similarly, the process 1→3, 2→2, 3→1, 4→4
will convert TONE to NOTE. When the process is applied again, NOTE returns to
TONE. (Not very interesting, is it?)

Your challenge begins with
the five letters A, E, M, S, and T. Use them to form a common English word.
Then, rearrange the letters to form a second common English word. Finally,
apply the same process of rearrangement to form a third common English word.

Can you do it?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesReason abstractly and quantitatively.Make sense of problems and persevere in solving them.CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2

ProblemsGrades: 9th to 12th, 3rd to 5th, 6th to 8thGeometryThe Number SystemMathematical PracticesExpressing Geometric Properties with EquationsGraph points on the coordinate plane to solve real-world and mathematical problems.Solve real-world and mathematical problems involving area, surface area, and volume.Apply and extend previous understandings of numbers to the system of rational numbers.Draw and identify lines and angles, and classify shapes by properties of their lines and angles.Make sense of problems and persevere in solving them.Understand and apply the Pythagorean Theorem.8.G.B.8, CCSS.Math.Practice.MP1, 4.G.A.1, 6.NS.C.8, 6.G.A.3, 5.G.A.1, HSG-GPE.B.6, HSG-GPE.B.7

Students in Mrs. Walker’s classroom had an estimation
contest. The student whose estimate is
the closest to the number of marbles in a jar wins the contest. Vicki, who
estimated 135 marbles, won the contest. Timothy, who estimated 150 marbles, got
second place. Lyon, who estimated 152, got third place. And Quinn, who
estimated 131, got fourth place. What is the exact number of marbles in the
jar?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesAttend to precision.Reason abstractly and quantitatively.Make sense of problems and persevere in solving them.CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6

Three lines cut rectangle ABCD into six congruent
(identical) squares.

The perimeter of rectangle ABCD is 30
cm. What is the area of the shaded region, in square centimeters?

ProblemsGrades: 9th to 12th, 6th to 8th, 3rd to 5thMathematical PracticesMeasurement & DataReason abstractly and quantitatively.Make sense of problems and persevere in solving them.Geometric measurement: understand concepts of area and relate area to multiplication and to addition.Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.4.MD.A.3, 3.MD.C.6, 3.MD.C.5a, 3.MD.C.5b, 3.MD.C.7a, 3.MD.C.7b, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2

Suppose you found an old roll of 15¢ stamps. Can you use a combination of 33¢ stamps and 15¢ stamps to mail a package for exactly $1.77?

ProblemsGrades: 6th to 8th, 3rd to 5th, 9th to 12thRatio & ProportionNum & Ops Base TenMathematical PracticesAlgebraic ThinkingUnderstand ratio concepts and use ratio reasoning to solve problems.Use place value understanding and properties of operations to perform multi-digit arithmetic.Look for and make use of structure.Attend to precision.Reason abstractly and quantitatively.Make sense of problems and persevere in solving them.Multiply and divide within 100.3.OA.C.7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7, 4.NBT.B.5, 6.RP.A.3a